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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"2c7966a7fdd5e4bfbdd722d24c36ab407f9a2d57e132b873c513b8941f810e35","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2ee8dc15b8ca5b2d","role":"sequence","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2ee8dc15b8ca5b2d","role":"sequence","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   12,     88,      152,         48;\n   2:   24,    328,     1296,       1584,         384;\n   3:   48,   1056,     7552,      20736,       19584,         3840;\n   4:   96,   3104,    35200,     172800,      364800,       280320,         46080;\n   5:  192,   8576,   141696,    1094400,     4108800,      7050240,       4561920,         645120;\n   6:  384,  22656,   514304,    5757696,    33868800,    103111680,     149022720,       83220480,       10321920;\n   7:  768,  57856,  1728512,   26492928,   225189888,   1071759360,    2745630720,     3426877440,     1682472960,      185794560;\n   8: 1536, 143872,  5474304,  110149632,  1278885888,   8778792960,   35115171840,    77662126080,    85279703040,    37344706560,     3715891200;\n   9: 3072, 350208, 16537600,  423014400,  6436454400,  60259368960,  347250032640,  1198374912000,  2331721728000,  2285273088000,   902961561600,    81749606400;\n  10: 6144, 837632, 48074752, 1524019200, 29439590400, 360768798720, 2832623861760, 14073194741760, 42714169344000, 74187767808000, 65644933939200, 23625636249600, 1961990553600;\n","recurrence_sentence":"The row polynomials satisfy (3+(4*n+2)*x+(4*n^2+8*n+4)*x^2)*P(n,x) = (6+(20*n+18)*x+(24*n^2+48*n+20)*x^2+(16*n^3+56*n^2+64*n+24)*x^3)*P(n-1,x) - (6*x^2+(8*n+12)*x^3+(8*n^2+16*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 12 + 88*x + 152*x^2 + 48*x^3.","mathematica":"P[1, x_] := 12 + 88*x + 152*x^2 + 48*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*P[n - 1, x] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*D[P[n - 1, x], x])/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 12 + 88*x + 152*x^2 + 48*x^3; for(n = 2, N, my(i = n); v[i] = ((6 + (20*n + 18)*x + (24*n^2 + 48*n + 20)*x^2 + (16*n^3 + 56*n^2 + 64*n + 24)*x^3)*v[i - 1] - (6*x^2 + (8*n + 12)*x^3 + (8*n^2 + 16*n + 8)*x^4)*deriv(v[i - 1], x))/(3 + (4*n + 2)*x + (4*n^2 + 8*n + 4)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f9379b86f8cefc86"],"text":"A395972 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A395972 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}